55.26.11 problem 11

Internal problem ID [13758]
Book : Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second edition
Section : Chapter 1, section 1.4. Equations Containing Polynomial Functions of y. subsection 1.4.1-2 Abel equations of the first kind.
Problem number : 11
Date solved : Thursday, October 02, 2025 at 07:58:19 AM
CAS classification : [_Abel]

\begin{align*} y^{\prime }&=a \,x^{n} y^{3}+3 a b \,x^{n +m} y^{2}+c \,x^{k} y-2 a \,b^{3} x^{n +3 m}+b c \,x^{m +k}-b m \,x^{m -1} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 219

ode:=diff(y(x),x) = a*x^n*y(x)^3+3*a*b*x^(m+n)*y(x)^2+c*x^k*y(x)-2*a*b^3*x^(n+3*m)+b*c*x^(m+k)-b*m*x^(m-1); 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= -x^{m} b \\ y &= -\frac {{\mathrm e}^{x \left (\frac {c \,x^{k}}{k +1}-\frac {3 a \,x^{n +2 m} b^{2}}{2 m +n +1}\right )}}{\sqrt {c_1 -2 a \int x^{n} {\mathrm e}^{-\frac {6 x \left (a \,x^{n +2 m} b^{2} \left (k +1\right )-\frac {2 x^{k} \left (m +\frac {n}{2}+\frac {1}{2}\right ) c}{3}\right )}{\left (k +1\right ) \left (2 m +n +1\right )}}d x}}-x^{m} b \\ y &= \frac {{\mathrm e}^{x \left (\frac {c \,x^{k}}{k +1}-\frac {3 a \,x^{n +2 m} b^{2}}{2 m +n +1}\right )}}{\sqrt {c_1 -2 a \int x^{n} {\mathrm e}^{-\frac {6 x \left (a \,x^{n +2 m} b^{2} \left (k +1\right )-\frac {2 x^{k} \left (m +\frac {n}{2}+\frac {1}{2}\right ) c}{3}\right )}{\left (k +1\right ) \left (2 m +n +1\right )}}d x}}-x^{m} b \\ \end{align*}

✓ Mathematica. Time used: 3.064 (sec). Leaf size: 232

ode=D[y[x],x]==a*x^n*y[x]^3+3*a*b*x^(n+m)*y[x]^2+c*x^k*y[x]-2*a*b^3*x^(n+3*m)+b*c*x^(m+k)-b*m*x^(m-1); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -b x^m-\frac {\exp \left (x \left (\frac {c x^k}{k+1}-\frac {3 a b^2 x^{2 m+n}}{2 m+n+1}\right )\right )}{\sqrt {-2 \int _1^xa \exp \left (\frac {2 c K[1]^{k+1}}{k+1}-\frac {6 a b^2 K[1]^{2 m+n+1}}{2 m+n+1}\right ) K[1]^ndK[1]+c_1}}\\ y(x)&\to -b x^m+\frac {\exp \left (x \left (\frac {c x^k}{k+1}-\frac {3 a b^2 x^{2 m+n}}{2 m+n+1}\right )\right )}{\sqrt {-2 \int _1^xa \exp \left (\frac {2 c K[1]^{k+1}}{k+1}-\frac {6 a b^2 K[1]^{2 m+n+1}}{2 m+n+1}\right ) K[1]^ndK[1]+c_1}}\\ y(x)&\to -b x^m \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
n = symbols("n") 
b = symbols("b") 
m = symbols("m") 
c = symbols("c") 
k = symbols("k") 
y = Function("y") 
ode = Eq(2*a*b**3*x**(3*m + n) - 3*a*b*x**(m + n)*y(x)**2 - a*x**n*y(x)**3 - b*c*x**(k + m) + b*m*x**(m - 1) - c*x**k*y(x) + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out